Higher-order averaging schemes in systems with fast and slow phases

Non-linear oscillatory systems containing a fast phase and a relatively slow phase are considered. A modified averaging method is proposed for the situation in which the slow variables, averaged over the fast phase, do not change. A procedure for separating the variables over substantially longer time intervals with respect to a small parameter is proposed and justified; over these extended time intervals, all the variables experience considerable evolution. Examples illustrating the efficiency of the proposed approach are presented. (C) 2002 Elsevier Science Ltd. All rights reserved.